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Easy Multiplication Products

This related thread quantifies the number-theoretical relationships of products in a multiplication table to the number base r. The previous, geeky thread laid out the ultimate objective for such a study: quantify the ease of memorization of a given table. I recognized in that thread that there are two main components in the problem; human cognitive ability and elementary number theory, that the latter seems “less difficult” to quantify. The multiplication-order thread addresses number theory, but it is clear that it doesn’t say everything about how easy it might be to memorize a multiplication table.

I am an American who learned multiplication in 2nd and 3rd grade (ages 7-8, between 1977 and 1978). In that era, rote memorization of product lines was the rule. We would memorize “the fives”, i.e., the line of products with five as the multiplier: {5, 10, 15, …, 50}, at first through 10 as the multiplicand. We then learned the elevens and twelves, so that ultimately we memorized a table with 78 unique products, rather than the strictly-decimal table of 55 unique products. This is because the eleven product line is nearly trivial {11, 22, 33, …, 99, 110, 121, 132}, and the dozens are, as my aunt put it “really very important, twelve is a very useful number”.

This method of quantifying “easily-memorized” facts will use the following definitions. We will not stick to the number-theoretical classifications of digits and relationships with base r, though these are important because we can calculate the “ease” using these. We will need to presume some things about human cognition, because I am not a researcher / cognitive scientist, and don’t have the benefit of a university with grants, etc. to build a proper study. (But this would be a cool thing to study, if it might be studied). We need some basic definitions so we can talk about the subject. Let the integer r ≥ 2 be a number base. Let the non-negative integer n < r be a digit of base r, with the digit “0” representing congruence with r. (We will ignore the special case where “0” signifies actual zero, i.e., when the place-value “0” stands alone.) Here are the “rules” of the quantification of “ease” of memorization of a multiplication table in base r:

We need two metrics. The first is number-theoretical, but can be observed without resorting to theory (however I have a way to calculate these and may post it later). The second relates to human cognitive ability and is based on a notion I think would govern how much a person can keep in mind while calculating. Yes, it’s a guess, and the value of this number would alter the calculations.

Let the positive rational number λ be the “period length“ of the cycle of unit-place-values in a product line for digit n in base r. I will assume that product lines of any digit n of base r with an integer λ, whether the unit-place-values increase or decrease with an increasing multiplicand, will be “easy” to memorize. Examples include digit 2 base 10 {0, 2, 4, 6, 8, 10, 12, …} and digit 8 base 10 {8, 16, 24, 32, 40, 48, …}. I will assume that any product line of any digit n of base r with a non-integer λ will be opaque to pupils, and not be “easy” to memorize. Example of this case is digit 4 base 10: {4, 8, 12, 16, 20, …}. For digit n = 2 of base r = 10, λ = 5. For digit n = 4 of base r = 12, λ = 3. (For any n such that n | r, λ = r/n.)

Let the positive rational number μ be the “mnemonic capacity“ of an average human being. We will suppose this is equivalent to the maximum subitizing range of 7. (For more about subitizing, search it, or take a look at this neat test at Wolfram). The human ability to subitize, that is, reliably quantify without counting, falls between 4 and 7 and a cogent study would be in order to really define this variable. I use 7 because this length had served the longest local telephone number in the United States, studied by the American phone monopoly a while back. A more solid value could be 6; for some it might be 4. Let’s presume μ = 7.

If any product line has λ > μ, we will deem it “difficult” to memorize. Conversely, any product line with λ ≤ μ will be deemed “easy” to memorize.

We will universally declare any product involving the trivial divisors {1, r}, common to all bases r, trivial to memorize. The problems “1 × x” and “10 × x” would seem to present little challenge to compute, no matter the size of the number base r.

We will universally presume any product involving the largest digit, ω = (r − 1), “easy” to memorize.

Easily Memorized Product Maps

Here are a few “easily memorized product maps” in several small bases r that we always find ourselves talking about. Since we presume μ = 7, we won’t analyze bases less than 8. Let’s look at maps for bases {8, 10, 12, 14, 15, 16}. In the maps, purple signifies “trivial products”, red “easy products”, and muted colors “difficult products”. The “difficult” beige and the gray products represent those that are regular or semi-coprime, and those that are coprime, respectively.

Quantifying the Ease of Memorization of Multiplication Tables

Let the integer e be the number of qualifying digits in base r. Let Me be the quantity of “easily-memorized products” in base r:

Me = er − Tri(e − 1)

Let the positive integer M be the total population of unique products in the multiplication table of base r:

M = Tri(r)

Now we can produce the following table for the bases considered:

r

e

M

Me

Me, %

Set E

8

6

36

33

91.7

{0, 1, 2, 4, 6, 7}

9

5

45

35

77.8

{0, 1, 3, 6, 8}

10

6

55

45

81.8

{0, 1, 2, 5, 8, 9}

11

3

66

30

45.5

{0, 1, a}

12

10

78

75

96.2

{0, 1, 2, 3, 4, 6, 8, 9, a, b}

13

3

91

36

39.5

{0, 1, c}

14

6

105

69

65.7

{0, 1, 2, 7, c, d}

15

7

120

84

70.0

{0, 1, 3, 5, a, c, e}

16

8

136

91

66.9

{0, 1, 2, 4, 8, c, e, f}

17

3

153

48

31.4

{0, 1, g}

18

8

171

116

67.8

{0, 1, 3, 6, 9, c, f, h}

19

3

190

54

28.4

{0, 1, j}

20

8

210

132

62.9

{0, 1, 4, 5, a, f, g, j}

Thus, it is clear that the decimal multiplication table is rather “easy to memorize”. Only dozenal and octal exceed the “ease of memorization” of the decimal multiplication table for r > 7. (All r ≤ 7 are trivial to memorize.) The hexadecimal table is significantly more difficult to memorize, on par with the tables for bases {14, 15, 18}.

Further, the computations suggest that the duodecimal multiplication table is the easiest table to memorize outside of bases r < 7. It is also the largest of {2, 3, 4, 5, 6, 7, 8, 10, 12}, meaning that we maximize our human computational capability when we use base twelve. With only three “difficult” products to memorize, a case could be made that the three holdouts could be tackled in a lesson, overcoming the “difficulty”.

All this is written based on an assumption; thus if the actual value of μ is different, or if we deem sequences like that of digit 4 base 10 {0, 4, 8, 2, 6} “easily memorized”, or the multiplication facts associated with the multiplier 2 in any base as “easily memorized”, our results will be different. (Note that in the table above, I gave digits 2 and e base 16 the “benefit of the doubt”, and included them in the total, though their values of μ disqualify them.)

I would argue for taking μ = 5: this supports what I got when I tried the subitizing game you linked to, and also the results shown on the page (with a steep drop between 5 and 6 objects).

Here's a table for bases 6 ≤ r ≤ 20, as well as 6:10 and 12:10, presuming μ = 5.

In this case there are so many two-digit multipliers in the table with reasonably long lines (like a full 12-times and an almost-full 15-times table, as well as a half-full 24-times table) that I fear it may not be accurate to quantify their difficulty just by their last digit alone. Assuming both digits must be remembered, then only the lines with length shorter than 6, as well as 10 and 20 for being copies of the 1 and 2 lines, can be considered easy, while the lines of 12 and 15 are not. (But 24 is as its length is only 5.)

r

e

M

Me

Me, %

Set E

6

6

21

21

100.0

{0, 1, 2, 3, 4, 5}

7

3

28

18

64.3

{0, 1, 6}

8

6

36

33

91.7

{0, 1, 2, 4, 6, 7}

9

5

45

35

77.8

{0, 1, 3, 6, 8}

10

6

55

45

81.8

{0, 1, 2, 5, 8, 9}

6:10

11

66

56

84.8

{0, 1, 2, 5, 8, 9, 10, 12, 15, 20, 30}

11

3

66

30

45.5

{0, 1, a}

12

8

78

68

87.2

{0, 1, 3, 4, 6, 8, 9, b}

13

3

91

36

39.5

{0, 1, c}

12:10

11

96

66

68.8

{0, 1, 2, 5, 8, 9, 10, 20, 24, 30, 60}

14

4

105

50

47.6

{0, 1, 7, d}

15

7

120

84

70.0

{0, 1, 3, 5, a, c, e}

16

6

136

81

59.6

{0, 1, 4, 8, c, f}

17

3

153

48

31.4

{0, 1, g}

18

6

171

93

54.4

{0, 1, 6, 9, c, h}

19

3

190

54

28.4

{0, 1, i}

20

8

210

132

62.9

{0, 1, 4, 5, a, f, g, j}

Tetradecimal and hexadecimal look really sad now! Now octal beats dozenal slightly, due to size being a more important factor: it is in fractions that octal fails. Similarly, it is in efficiency that senary fails. Neither of these factors are covered in this study. This seems to cut the easy-to-memorize tables to {2, 3, 4, 5, 6, 8, 10, 12, 6:10}, presuming we need Me ≥ 80% for a multiplication table to be a viable option for memorization. (This measure does cut out nonary, which is far closer to 80% than any of the other runner-ups. Nevertheless, I think nonary's out because it's odd: 2 is so important that having a difficult-to-memorise line for 2 really cripples a base.)

Given what we've been talking about recently at various threads, such as this one, we should try μ = 9 or 10. Now everything up to nonary or decimal is trivial, everything up to heptadecimal (and enneadecimal) is the same, but with μ = 9, octodecimal now has {0, 1, 2, 3, 6, 9, c, f, g, h} easy, making a figure of 78.9% (171 - 36 = 135), and with μ = 10, vigesimal has {0, 1, 2, 4, 5, a, f, g, i, j} easy, making a figure of 73.8% (210 - 55 = 155).

Tetravigesimal still has only {0, 1, 3, 4, 6, c, i, k, l, n} being helpful, resulting in a much lower figure of 65.0% (300 - 105 = 195). I am not sure how to measure the length of the line as a factor, though. It seems clear that past 20 this is a serious problem.